Subgroups — C₂×C₃×ΓL(1,𝔽₈)⋊D₆₇ #30683192

Order 16884.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×ΓL(1,𝔽₈)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×ΓL(1,𝔽₈)⋊D₆₇	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×D₄₆₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×ΓL(1,𝔽₈)⋊D₆₇	C₃	1	3	yes	maximal, direct product factor
5	C₃×ΓL(1,𝔽₈)×C₆₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	ΓL(1,𝔽₈)⋊D₆₇	C₂×C₃	1	6	yes	direct product factor
7	C₃×D₄₆₉	C₂×C₃	1	2	yes	complemented
8	C₂×ΓL(1,𝔽₈)×C₆₇	C₂×C₃	1	3	yes	complemented
9	C₂×C₃×C₇×C₆₇	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₃²×D₆₇	—	7	1	no	maximal, Hall
11	C₂×D₄₆₉	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×C₇×C₆₇	C₂²×C₃	1	1	yes	characteristic, complemented
13	ΓL(1,𝔽₈)×C₆₇	C₂²×C₃	1	3	yes	complemented
14	C₂×C₃²×C₆₇	—	7	1	no	—
15	C₃²×D₆₇	—	7	2	no	—
16	C₂×C₇×C₆₇	C₂×C₃²	1	1	yes	characteristic, complemented
17	D₄₆₉	C₂×C₃²	1	2	yes	complemented
18	C₂×C₃×D₆₇	—	7	3	no	—
19	C₂×C₃×D₆₇	—	21	1	no	—
20	C₃²×C₆₇	—	7	1	no	Hall
21	C₇×C₆₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃×C₆₇	—	7	3	no	—
23	C₂×C₃×C₆₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
24	C₃×D₆₇	—	7	6	no	—
25	C₃×D₆₇	—	21	2	no	—
26	C₂×D₆₇	—	21	1	no	Hall
27	C₂×C₃×AGL(1,𝔽₇)	—	67	1	no	maximal, Hall
28	C₃×C₆₇	—	7	3	no	—
29	C₃×C₆₇	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
30	C₂×C₆₇	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
31	D₆₇	—	21	2	no	—
32	C₂×C₃×ΓL(1,𝔽₈)	D₆₇	1	1	yes	characteristic, complemented
33	C₃×AGL(1,𝔽₇)	—	67	2	no	—
34	C₂×C₃×D₇	—	67	1	no	—
35	C₂×AGL(1,𝔽₇)	—	67	3	no	—
36	C₆₇	C₂×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
37	C₃×ΓL(1,𝔽₈)	C₂×D₆₇	1	1	yes	characteristic, complemented, Hall
38	C₂×C₃×C₇	C₃×D₆₇	1	1	yes	characteristic, complemented
39	C₃×D₇	—	67	2	no	—
40	C₂×ΓL(1,𝔽₈)	C₃×D₆₇	1	3	yes	complemented
41	AGL(1,𝔽₇)	—	67	6	no	—
42	C₂²×C₃²	—	469	1	no	Hall
43	C₂×D₇	—	67	1	no	Hall
44	C₃×C₇	C₂×C₃×D₆₇	1	1	yes	characteristic, complemented
45	ΓL(1,𝔽₈)	C₂×C₃×D₆₇	1	3	yes	complemented
46	C₂×C₃²	—	7	1	no	—
47	C₂×C₃²	—	469	2	no	—
48	C₂×C₇	C₃²×D₆₇	1	1	yes	characteristic, complemented
49	D₇	—	67	2	no	—
50	C₂²×C₃	—	469	3	no	—
51	C₂²×C₃	—	469	1	no	—
52	C₃²	—	7	1	no	Sylow
53	C₇	C₂×C₃²×D₆₇	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₃	—	7	3	no	—
55	C₂×C₃	—	1407	2	no	—
56	C₂×C₃	—	469	6	no	—
57	C₂×C₃	ΓL(1,𝔽₈)⋊D₆₇	1	1	yes	characteristic, direct product factor
58	C₂²	—	469	1	no	Sylow
59	C₃	—	7	3	no	—
60	C₃	C₂×ΓL(1,𝔽₈)⋊D₆₇	1	1	yes	characteristic, direct product factor
61	C₂	—	1407	2	no	—
62	C₂	C₃×ΓL(1,𝔽₈)⋊D₆₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	ΓL(1,𝔽₈)⋊D₆₇
commutator, cosocle, hypocenter	C₇×C₆₇	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇×C₆₇	C₂×C₃
LCS	G ▷ C₇×C₆₇
UCS	C₁ ◁ C₂×C₃

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃×ΓL(1,𝔽₈)×C₆₇	C₂	semidirect
2. C₃×ΓL(1,𝔽₈)⋊D₆₇	C₂	direct (N × H)
3. C₂×C₃×D₄₆₉	C₃	semidirect
4. C₂×ΓL(1,𝔽₈)⋊D₆₇	C₃	direct (N × H)
5. C₃×ΓL(1,𝔽₈)×C₆₇	C₂²	semidirect
6. ΓL(1,𝔽₈)⋊D₆₇	C₂×C₃	direct (N × H)
7. C₃×D₄₆₉	C₂×C₃	semidirect
8. C₂×ΓL(1,𝔽₈)×C₆₇	C₂×C₃	semidirect
9. C₂×C₃×C₇×C₆₇	C₂×C₃	semidirect
11. C₂×D₄₆₉	C₃²	semidirect
12. C₃×C₇×C₆₇	C₂²×C₃	semidirect
13. ΓL(1,𝔽₈)×C₆₇	C₂²×C₃	semidirect
16. C₂×C₇×C₆₇	C₂×C₃²	semidirect
17. D₄₆₉	C₂×C₃²	semidirect
21. C₇×C₆₇	C₂²×C₃²	semidirect
23. C₂×C₃×C₆₇	AGL(1,𝔽₇)	semidirect
29. C₃×C₆₇	C₂×AGL(1,𝔽₇)	semidirect
30. C₂×C₆₇	C₃×AGL(1,𝔽₇)	semidirect
32. C₂×C₃×ΓL(1,𝔽₈)	D₆₇	semidirect
36. C₆₇	C₂×C₃×AGL(1,𝔽₇)	semidirect
37. C₃×ΓL(1,𝔽₈)	C₂×D₆₇	semidirect
38. C₂×C₃×C₇	C₃×D₆₇	semidirect
40. C₂×ΓL(1,𝔽₈)	C₃×D₆₇	semidirect
44. C₃×C₇	C₂×C₃×D₆₇	semidirect
45. ΓL(1,𝔽₈)	C₂×C₃×D₆₇	semidirect
48. C₂×C₇	C₃²×D₆₇	semidirect
53. C₇	C₂×C₃²×D₆₇	semidirect
57. C₂×C₃	ΓL(1,𝔽₈)⋊D₆₇	direct (N × H)
60. C₃	C₂×ΓL(1,𝔽₈)⋊D₆₇	direct (N × H)
62. C₂	C₃×ΓL(1,𝔽₈)⋊D₆₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×ΓL(1,𝔽₈)×C₆₇	2
C₃×ΓL(1,𝔽₈)⋊D₆₇	2
C₂×C₃×D₄₆₉	3
C₂×ΓL(1,𝔽₈)⋊D₆₇	3
C₂×C₃²×D₆₇	7
C₂×C₃×AGL(1,𝔽₇)	67

Subgroups — C2×C3×ΓL(1,𝔽8)⋊D67 #30683192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×ΓL(1,𝔽₈)⋊D₆₇ #30683192